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Transcript
F.H.R.
1
Trigonometric functions
The general expression for angles with a given trigonometric ratio:
The topic of this lesson is functions in which the trigonometric ratios of the
unknown quantity occur. Equations of this type may be reduced to the
solution of one or more equations of the type:
Sin x = , Cos x =  or Tan x = 
In this kind of equation,  is known and x is the unknown element.
Example 1.
Consider the equation
Tan x = 1
One solution is Tan x = 45 but that is not the only solution because it is a
fact that:
(Tan  = 45 + 180 ) = Tan 45 = 1
 = 225 is another solution to this equation. The general solution is:
 = 45 + n 180 where n is an integer, either negative, 0 or positive. By
giving n different values, different solutions to the equation are obtained.
Figure 1.
Graph of y = Tan x
F.H.R.
2
Reference to figure 1 may help showing this. The horizontal line y =  is
drawn on the same scale and intersects the tangent curve in the points A,
A’, A”. B, B’ etc. If the absissa of A = , that of A’ is  + 180,
of A” =  + 2 x 180
For the equation Cos  =  it must be realized that if  is numerically
greater than 1, the equation will have no solution. When  is numerically less
than 1, follow this procedure: figure 2 shows the graph of y = cos  and the
line y =  is drawn on the same scale. The abscissae of the point of
intersection will give the solution to the equation Cos  = 
Figure 2.
Graph of y = cos x
If  is the smallest angle for which y = cos  the absicca of A, A’, A” etc
are 360 - , 360 + , etc. These are all particular cases of the formula:
x = n360 - ,
In this equation n is an integer, positive, 0 or negative.
The procedure for Sin x =  is the same as cos x = 
The general solution to the equation Sin x =  is
x = n180 + (-1)n ,
n is an integer, positive, 0 or negative.
F.H.R.
Figure 3.
3
Graph of y = sin x
Example 2
Find the general solution to the equation Sin x = 0.515 which lay in the range
0 to 360 state the general solution
Solution:
From your calculators you will find that Sin 31 = 0.515, therefore 31 is a
solution. The general solution is however: x = n 180 + (-1)n 31. With n = 0 or
1 the solution in the range 0 to 360 is obtained. That is 31 and
180 - 31 = 149